One brief observation based on watching the John Adams miniseries... our modern day Republicans should probably be called Federalists (the party of Adams, not Jefferson), whereas the Democrats are descended from Jefferson's original Democratic-Republican Party. However, neither party really seems to be Republican in nature, as Jefferson, I think, would define. Neither party seems to be truly in support of small central government and strong States' rights. The only connection that is really obvious is that the Neoconservative Movement that has steamrolled the GOP (aka the "Republicans") is for a strong federal government, a strong executive branch, and a strong standing military. All of these attributes were shared by the Federalists of Alexander Hamilton.
One thing is clear: both parties probably need to change their names soon to something more accurate, and perhaps more representative of the will of the People. Libertarians haven't been overly successful, nor has the Green Party. Perhaps we'll see a Populist Party rise some day, or maybe even a return of the Democratic-Republican Party as a hybrid of both main parties.
References:
Federalist Party (United States)
Federalists and Anti-Federalists
Democratic-Republican Party
Republican Party (United States)
